an integrated solver for optimization problemscoral.ie.lehigh.edu/mip-2006/talks/yunes.pdf · an...
TRANSCRIPT
![Page 1: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/1.jpg)
An Integrated Solver for Optimization Problems
Ionut D. Aron1 John N. Hooker2 Tallys H. Yunes3
1Department of Computer Science, Brown University
2Tepper School of Business, Carnegie Mellon University
3Department of Management Science, University of Miami
June 6 2006
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I’d Like To Thank the Program Committee for...
Inviting me to be here
Placing my talk right after Jon Lee’s
I A “foolish” model depends on your vocabularyI Larger vocabulary ⇒ more natural modelsI Some things we once worried about are now automatic
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 2
![Page 3: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/3.jpg)
I’d Like To Thank the Program Committee for...
Inviting me to be here
Placing my talk right after Jon Lee’s
I A “foolish” model depends on your vocabularyI Larger vocabulary ⇒ more natural modelsI Some things we once worried about are now automatic
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 2
![Page 4: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/4.jpg)
I’d Like To Thank the Program Committee for...
Inviting me to be here
Placing my talk right after Jon Lee’s
I A “foolish” model depends on your vocabularyI Larger vocabulary ⇒ more natural modelsI Some things we once worried about are now automatic
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 2
![Page 5: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/5.jpg)
I’d Like To Thank the Program Committee for...
Inviting me to be here
Placing my talk right after Jon Lee’sI A “foolish” model depends on your vocabulary
I Larger vocabulary ⇒ more natural modelsI Some things we once worried about are now automatic
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 2
![Page 6: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/6.jpg)
I’d Like To Thank the Program Committee for...
Inviting me to be here
Placing my talk right after Jon Lee’sI A “foolish” model depends on your vocabularyI Larger vocabulary ⇒ more natural models
I Some things we once worried about are now automatic
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 2
![Page 7: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/7.jpg)
I’d Like To Thank the Program Committee for...
Inviting me to be here
Placing my talk right after Jon Lee’sI A “foolish” model depends on your vocabularyI Larger vocabulary ⇒ more natural modelsI Some things we once worried about are now automatic
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 2
![Page 8: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/8.jpg)
Outline
Introduction and Motivation
SIMPL Concepts
3 Modeling Examples
Computational Experiments
Future Work and Conclusion
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 3
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Why Integrate?
Integration: combination of two or more solution techniquesinto a well-coordinated optimization algorithm
Recent research shows integration can sometimes significantlyoutperform traditional methods in
I Planning and scheduling (jobs, crews, sports, etc.)I Routing and transportationI Engineering and network designI ManufacturingI Inventory managementI Etc.
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 4
![Page 10: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/10.jpg)
Why Integrate?
Integration: combination of two or more solution techniquesinto a well-coordinated optimization algorithm
Recent research shows integration can sometimes significantlyoutperform traditional methods in
I Planning and scheduling (jobs, crews, sports, etc.)I Routing and transportationI Engineering and network designI ManufacturingI Inventory managementI Etc.
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 4
![Page 11: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/11.jpg)
Why Integrate?
Integration: combination of two or more solution techniquesinto a well-coordinated optimization algorithm
Recent research shows integration can sometimes significantlyoutperform traditional methods in
I Planning and scheduling (jobs, crews, sports, etc.)I Routing and transportationI Engineering and network designI ManufacturingI Inventory managementI Etc.
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 4
![Page 12: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/12.jpg)
Why Integrate?
Integration: combination of two or more solution techniquesinto a well-coordinated optimization algorithm
Recent research shows integration can sometimes significantlyoutperform traditional methods in
I Planning and scheduling (jobs, crews, sports, etc.)I Routing and transportationI Engineering and network designI ManufacturingI Inventory managementI Etc.
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 4
![Page 13: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/13.jpg)
Integration Continued
We are mostly interested in combining traditional OR techniques(LP, MILP) with Constraint Programming
Some benefits of integration:
Models are simpler, smaller and more natural
It combines complementary strengths of different optimizationtechniques
Problem structure is more easily captured and exploited
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 5
![Page 14: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/14.jpg)
Integration Continued
We are mostly interested in combining traditional OR techniques(LP, MILP) with Constraint Programming
Some benefits of integration:
Models are simpler, smaller and more natural
It combines complementary strengths of different optimizationtechniques
Problem structure is more easily captured and exploited
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 5
![Page 15: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/15.jpg)
Integration Continued
We are mostly interested in combining traditional OR techniques(LP, MILP) with Constraint Programming
Some benefits of integration:
Models are simpler, smaller and more natural
It combines complementary strengths of different optimizationtechniques
Problem structure is more easily captured and exploited
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 5
![Page 16: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/16.jpg)
Integration Continued
We are mostly interested in combining traditional OR techniques(LP, MILP) with Constraint Programming
Some benefits of integration:
Models are simpler, smaller and more natural
It combines complementary strengths of different optimizationtechniques
Problem structure is more easily captured and exploited
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 5
![Page 17: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/17.jpg)
Integration Continued
We are mostly interested in combining traditional OR techniques(LP, MILP) with Constraint Programming
Some benefits of integration:
Models are simpler, smaller and more natural
It combines complementary strengths of different optimizationtechniques
Problem structure is more easily captured and exploited
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 5
![Page 18: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/18.jpg)
Integration Continued
We are mostly interested in combining traditional OR techniques(LP, MILP) with Constraint Programming
Some benefits of integration:
Models are simpler, smaller and more natural
It combines complementary strengths of different optimizationtechniques
Problem structure is more easily captured and exploited
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 5
![Page 19: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/19.jpg)
Constraint Programming (CP)
Originated from the AI and CS communities (80’s)
Concerned with Constraint Satisfaction Problems (CSPs):
I Given variables xi ∈ Di
I Given constraints ci : D1 × · · · ×Dn → {T, F}I Assign values to variables to satisfy all constraints
Main Ideas:
I Constraints eliminate infeasible values: domain reductionI Local inferences are shared: constraint propagation
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 6
![Page 20: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/20.jpg)
Constraint Programming (CP)
Originated from the AI and CS communities (80’s)
Concerned with Constraint Satisfaction Problems (CSPs):I Given variables xi ∈ Di
I Given constraints ci : D1 × · · · ×Dn → {T, F}I Assign values to variables to satisfy all constraints
Main Ideas:
I Constraints eliminate infeasible values: domain reductionI Local inferences are shared: constraint propagation
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 6
![Page 21: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/21.jpg)
Constraint Programming (CP)
Originated from the AI and CS communities (80’s)
Concerned with Constraint Satisfaction Problems (CSPs):I Given variables xi ∈ Di
I Given constraints ci : D1 × · · · ×Dn → {T, F}I Assign values to variables to satisfy all constraints
Main Ideas:I Constraints eliminate infeasible values: domain reductionI Local inferences are shared: constraint propagation
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 6
![Page 22: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/22.jpg)
Constraint Programming (CP)
Originated from the AI and CS communities (80’s)
Concerned with Constraint Satisfaction Problems (CSPs):I Given variables xi ∈ Di
I Given constraints ci : D1 × · · · ×Dn → {T, F}I Assign values to variables to satisfy all constraints
Main Ideas:I Constraints eliminate infeasible values: domain reductionI Local inferences are shared: constraint propagation
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 6
![Page 23: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/23.jpg)
Constraint Programming (continued)
When compared to MILP models, CP models usually have
More informative (larger) variable domains:
I city ∈ { Atlanta, Boston, Miami, San Francisco }I `i = location of facility iI contrast with: xij = 1 if facility i is placed in location j
More expressive constraints:
I alldifferent: all variables from a set assume distinct valuesI element: implements variable indices (Cx)I cumulative: job scheduling with resource constraintsI etc.I These are called global constraints
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 7
![Page 24: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/24.jpg)
Constraint Programming (continued)
When compared to MILP models, CP models usually have
More informative (larger) variable domains:
I city ∈ { Atlanta, Boston, Miami, San Francisco }I `i = location of facility iI contrast with: xij = 1 if facility i is placed in location j
More expressive constraints:
I alldifferent: all variables from a set assume distinct valuesI element: implements variable indices (Cx)I cumulative: job scheduling with resource constraintsI etc.I These are called global constraints
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 7
![Page 25: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/25.jpg)
Constraint Programming (continued)
When compared to MILP models, CP models usually have
More informative (larger) variable domains:I city ∈ { Atlanta, Boston, Miami, San Francisco }I `i = location of facility iI contrast with: xij = 1 if facility i is placed in location j
More expressive constraints:
I alldifferent: all variables from a set assume distinct valuesI element: implements variable indices (Cx)I cumulative: job scheduling with resource constraintsI etc.I These are called global constraints
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 7
![Page 26: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/26.jpg)
Constraint Programming (continued)
When compared to MILP models, CP models usually have
More informative (larger) variable domains:I city ∈ { Atlanta, Boston, Miami, San Francisco }I `i = location of facility iI contrast with: xij = 1 if facility i is placed in location j
More expressive constraints:I alldifferent: all variables from a set assume distinct valuesI element: implements variable indices (Cx)I cumulative: job scheduling with resource constraintsI etc.
I These are called global constraints
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 7
![Page 27: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/27.jpg)
Constraint Programming (continued)
When compared to MILP models, CP models usually have
More informative (larger) variable domains:I city ∈ { Atlanta, Boston, Miami, San Francisco }I `i = location of facility iI contrast with: xij = 1 if facility i is placed in location j
More expressive constraints:I alldifferent: all variables from a set assume distinct valuesI element: implements variable indices (Cx)I cumulative: job scheduling with resource constraintsI etc.I These are called global constraints
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 7
![Page 28: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/28.jpg)
Previous Work
Existing modeling languages and programming libraries supportintegration to a greater or lesser extent:
ECLiPSe, Rodosek, Wallace and Rajian 99
OPL, Van Hentenryck, Lustig, Michel and Puget 99
Mosel, Colombani and Heipcke 02
SCIP, Achterberg 04
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 8
![Page 29: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/29.jpg)
Previous Work
Existing modeling languages and programming libraries supportintegration to a greater or lesser extent:
ECLiPSe, Rodosek, Wallace and Rajian 99
OPL, Van Hentenryck, Lustig, Michel and Puget 99
Mosel, Colombani and Heipcke 02
SCIP, Achterberg 04
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 8
![Page 30: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/30.jpg)
Previous Work
Existing modeling languages and programming libraries supportintegration to a greater or lesser extent:
ECLiPSe, Rodosek, Wallace and Rajian 99
OPL, Van Hentenryck, Lustig, Michel and Puget 99
Mosel, Colombani and Heipcke 02
SCIP, Achterberg 04
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 8
![Page 31: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/31.jpg)
Previous Work (continued)
Some relevant concepts and techniques:
Allowing information exchange among solvers: Rodosek, Wallace &Hajian 99
Decomposition approaches: Benders 62, Eremin & Wallace 01,Hooker & Ottosson 03, Hooker & Yan 95, Jain & Grossmann 01
Relaxation of global constraints as systems of linear inequalities:Hooker 00, Refalo 00, Williams & Yan 01, Yunes 02
Propagation and variable fixing using relaxations: Focacci, Lodi &Milano 99
Generation of cutting planes as a form of logical inference:Bockmayr & Kasper 98, Bockmayr & Eisenbrand 00
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 9
![Page 32: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/32.jpg)
Previous Work (continued)
Some relevant concepts and techniques:
Allowing information exchange among solvers: Rodosek, Wallace &Hajian 99
Decomposition approaches: Benders 62, Eremin & Wallace 01,Hooker & Ottosson 03, Hooker & Yan 95, Jain & Grossmann 01
Relaxation of global constraints as systems of linear inequalities:Hooker 00, Refalo 00, Williams & Yan 01, Yunes 02
Propagation and variable fixing using relaxations: Focacci, Lodi &Milano 99
Generation of cutting planes as a form of logical inference:Bockmayr & Kasper 98, Bockmayr & Eisenbrand 00
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 9
![Page 33: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/33.jpg)
Previous Work (continued)
Some relevant concepts and techniques:
Allowing information exchange among solvers: Rodosek, Wallace &Hajian 99
Decomposition approaches: Benders 62, Eremin & Wallace 01,Hooker & Ottosson 03, Hooker & Yan 95, Jain & Grossmann 01
Relaxation of global constraints as systems of linear inequalities:Hooker 00, Refalo 00, Williams & Yan 01, Yunes 02
Propagation and variable fixing using relaxations: Focacci, Lodi &Milano 99
Generation of cutting planes as a form of logical inference:Bockmayr & Kasper 98, Bockmayr & Eisenbrand 00
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 9
![Page 34: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/34.jpg)
Previous Work (continued)
Some relevant concepts and techniques:
Allowing information exchange among solvers: Rodosek, Wallace &Hajian 99
Decomposition approaches: Benders 62, Eremin & Wallace 01,Hooker & Ottosson 03, Hooker & Yan 95, Jain & Grossmann 01
Relaxation of global constraints as systems of linear inequalities:Hooker 00, Refalo 00, Williams & Yan 01, Yunes 02
Propagation and variable fixing using relaxations: Focacci, Lodi &Milano 99
Generation of cutting planes as a form of logical inference:Bockmayr & Kasper 98, Bockmayr & Eisenbrand 00
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 9
![Page 35: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/35.jpg)
Previous Work (continued)
Some relevant concepts and techniques:
Allowing information exchange among solvers: Rodosek, Wallace &Hajian 99
Decomposition approaches: Benders 62, Eremin & Wallace 01,Hooker & Ottosson 03, Hooker & Yan 95, Jain & Grossmann 01
Relaxation of global constraints as systems of linear inequalities:Hooker 00, Refalo 00, Williams & Yan 01, Yunes 02
Propagation and variable fixing using relaxations: Focacci, Lodi &Milano 99
Generation of cutting planes as a form of logical inference:Bockmayr & Kasper 98, Bockmayr & Eisenbrand 00
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 9
![Page 36: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/36.jpg)
Previous Work (continued)
Some relevant concepts and techniques:
Allowing information exchange among solvers: Rodosek, Wallace &Hajian 99
Decomposition approaches: Benders 62, Eremin & Wallace 01,Hooker & Ottosson 03, Hooker & Yan 95, Jain & Grossmann 01
Relaxation of global constraints as systems of linear inequalities:Hooker 00, Refalo 00, Williams & Yan 01, Yunes 02
Propagation and variable fixing using relaxations: Focacci, Lodi &Milano 99
Generation of cutting planes as a form of logical inference:Bockmayr & Kasper 98, Bockmayr & Eisenbrand 00
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 9
![Page 37: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/37.jpg)
Previous Work (continued)
Some relevant concepts and techniques:
Allowing information exchange among solvers: Rodosek, Wallace &Hajian 99
Decomposition approaches: Benders 62, Eremin & Wallace 01,Hooker & Ottosson 03, Hooker & Yan 95, Jain & Grossmann 01
Relaxation of global constraints as systems of linear inequalities:Hooker 00, Refalo 00, Williams & Yan 01, Yunes 02
Propagation and variable fixing using relaxations: Focacci, Lodi &Milano 99
Generation of cutting planes as a form of logical inference:Bockmayr & Kasper 98, Bockmayr & Eisenbrand 00
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SIMPL Objectives
High-level modeling languageI Concise and easily understandable modelsI Natural specification of integrated modelsI Allow user to reveal problem structure to the solver
Low-level integrationI Increased effectiveness when underlying technologies interact
at a micro level during the search
Modularity, flexibility, extensibility, efficiencyI Make it easy to add new types of constraints, relaxations,
solvers and search strategies
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SIMPL Objectives
High-level modeling languageI Concise and easily understandable modelsI Natural specification of integrated modelsI Allow user to reveal problem structure to the solver
Low-level integrationI Increased effectiveness when underlying technologies interact
at a micro level during the search
Modularity, flexibility, extensibility, efficiencyI Make it easy to add new types of constraints, relaxations,
solvers and search strategies
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SIMPL Objectives
High-level modeling languageI Concise and easily understandable modelsI Natural specification of integrated modelsI Allow user to reveal problem structure to the solver
Low-level integrationI Increased effectiveness when underlying technologies interact
at a micro level during the search
Modularity, flexibility, extensibility, efficiencyI Make it easy to add new types of constraints, relaxations,
solvers and search strategies
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SIMPL Objectives
High-level modeling languageI Concise and easily understandable modelsI Natural specification of integrated modelsI Allow user to reveal problem structure to the solver
Low-level integrationI Increased effectiveness when underlying technologies interact
at a micro level during the search
Modularity, flexibility, extensibility, efficiencyI Make it easy to add new types of constraints, relaxations,
solvers and search strategies
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Main Idea Behind SIMPL
CP and MILP are special cases of a general method, ratherthan separate methods to be combined
Common solution strategy: Search-Infer-Relax
Search = enumeration of problem restrictions
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Main Idea Behind SIMPL
CP and MILP are special cases of a general method, ratherthan separate methods to be combined
Common solution strategy: Search-Infer-Relax
Search = enumeration of problem restrictions
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Main Idea Behind SIMPL
CP and MILP are special cases of a general method, ratherthan separate methods to be combined
Common solution strategy: Search-Infer-Relax
Search = enumeration of problem restrictions
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 11
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Main Idea Behind SIMPL
CP and MILP are special cases of a general method, ratherthan separate methods to be combined
Common solution strategy: Search-Infer-Relax
Search = enumeration of problem restrictions
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 11
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The Ubiquity of Search, Inference, Relaxation
SolutionMethod Restriction Inference Relaxation
MILP Branch onfractional vars.
Cutting planes,preprocessing
LP relaxation
CP Split variabledomains
Domain reduction,propagation
Current domains
CGO Split intervalsInterv. propag.,lagr. mult.
LP or NLP relaxation
Benders SubproblemBenders cuts(nogoods)
Master problem
DPL Branching Resolution andconfl. clauses
Processedconfl. clauses
Tabu SearchCurrentneighborhood Tabu list Same as restriction
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The Ubiquity of Search, Inference, Relaxation
SolutionMethod Restriction Inference Relaxation
MILP Branch onfractional vars.
Cutting planes,preprocessing
LP relaxation
CP Split variabledomains
Domain reduction,propagation
Current domains
CGO Split intervalsInterv. propag.,lagr. mult.
LP or NLP relaxation
Benders SubproblemBenders cuts(nogoods)
Master problem
DPL Branching Resolution andconfl. clauses
Processedconfl. clauses
Tabu SearchCurrentneighborhood Tabu list Same as restriction
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Constraint-Based Control
Search: constraints direct the searchI Each constraint has a branching moduleI This module creates new problem restrictions
Infer: constraints drive the inferenceI Each constraint has a filtering/inference moduleI This module creates new constraints to tighten the relaxations
Relax: constraints create the relaxationsI Each constraint has a relaxation moduleI This module reformulates the constraint according to different
relaxations (LP, MILP, CP, etc.)
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Constraint-Based Control
Search: constraints direct the searchI Each constraint has a branching moduleI This module creates new problem restrictions
Infer: constraints drive the inferenceI Each constraint has a filtering/inference moduleI This module creates new constraints to tighten the relaxations
Relax: constraints create the relaxationsI Each constraint has a relaxation moduleI This module reformulates the constraint according to different
relaxations (LP, MILP, CP, etc.)
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Constraint-Based Control
Search: constraints direct the searchI Each constraint has a branching moduleI This module creates new problem restrictions
Infer: constraints drive the inferenceI Each constraint has a filtering/inference moduleI This module creates new constraints to tighten the relaxations
Relax: constraints create the relaxationsI Each constraint has a relaxation moduleI This module reformulates the constraint according to different
relaxations (LP, MILP, CP, etc.)
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Constraint-Based Control
Search: constraints direct the searchI Each constraint has a branching moduleI This module creates new problem restrictions
Infer: constraints drive the inferenceI Each constraint has a filtering/inference moduleI This module creates new constraints to tighten the relaxations
Relax: constraints create the relaxationsI Each constraint has a relaxation moduleI This module reformulates the constraint according to different
relaxations (LP, MILP, CP, etc.)
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Example 1: Production Planning
Manufacture several products at a plant of limited capacity
Products made in one of several production modes (e.g. smallscale, medium scale, etc.)
x = quantity of a product
Only certain ranges of quantities are possible: gaps in thedomain of x
Net income function f(x) is semi-continuous piecewise linear
Objective: maximize net income
Previous work: Refalo (1999), Ottosson, Thorsteinsson andHooker (1999, 2002)
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Example 1: Production Planning
Manufacture several products at a plant of limited capacity
Products made in one of several production modes (e.g. smallscale, medium scale, etc.)
x = quantity of a product
Only certain ranges of quantities are possible: gaps in thedomain of x
Net income function f(x) is semi-continuous piecewise linear
Objective: maximize net income
Previous work: Refalo (1999), Ottosson, Thorsteinsson andHooker (1999, 2002)
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Example 1: Production Planning
Manufacture several products at a plant of limited capacity
Products made in one of several production modes (e.g. smallscale, medium scale, etc.)
x = quantity of a product
Only certain ranges of quantities are possible: gaps in thedomain of x
Net income function f(x) is semi-continuous piecewise linear
Objective: maximize net income
Previous work: Refalo (1999), Ottosson, Thorsteinsson andHooker (1999, 2002)
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 14
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Example 1: Production Planning
Manufacture several products at a plant of limited capacity
Products made in one of several production modes (e.g. smallscale, medium scale, etc.)
x = quantity of a product
Only certain ranges of quantities are possible: gaps in thedomain of x
Net income function f(x) is semi-continuous piecewise linear
Objective: maximize net income
Previous work: Refalo (1999), Ottosson, Thorsteinsson andHooker (1999, 2002)
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 14
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Example 1: Production Planning
Manufacture several products at a plant of limited capacity
Products made in one of several production modes (e.g. smallscale, medium scale, etc.)
x = quantity of a product
Only certain ranges of quantities are possible: gaps in thedomain of x
Net income function f(x) is semi-continuous piecewise linear
Objective: maximize net income
Previous work: Refalo (1999), Ottosson, Thorsteinsson andHooker (1999, 2002)
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 14
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Example 1: Production Planning
Manufacture several products at a plant of limited capacity
Products made in one of several production modes (e.g. smallscale, medium scale, etc.)
x = quantity of a product
Only certain ranges of quantities are possible: gaps in thedomain of x
Net income function f(x) is semi-continuous piecewise linear
Objective: maximize net income
Previous work: Refalo (1999), Ottosson, Thorsteinsson andHooker (1999, 2002)
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 14
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Example 1: Production Planning
Manufacture several products at a plant of limited capacity
Products made in one of several production modes (e.g. smallscale, medium scale, etc.)
x = quantity of a product
Only certain ranges of quantities are possible: gaps in thedomain of x
Net income function f(x) is semi-continuous piecewise linear
Objective: maximize net income
Previous work: Refalo (1999), Ottosson, Thorsteinsson andHooker (1999, 2002)
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 14
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Example 1: Production Planning
Manufacture several products at a plant of limited capacity
Products made in one of several production modes (e.g. smallscale, medium scale, etc.)
x = quantity of a product
Only certain ranges of quantities are possible: gaps in thedomain of x
Net income function f(x) is semi-continuous piecewise linear
Objective: maximize net income
Previous work: Refalo (1999), Ottosson, Thorsteinsson andHooker (1999, 2002)
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Example 1: Production PlanningShape of Net Income Function f(x)
f(x)
x
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Example 1: Production Planning: MILP
xi = quantity of product i (continuous)
yik = whether or not product i is made in mode k (binary)
λik, µik = weights for mode k (convex combination)
MILP Model
max∑
ik λikcik + µikdik∑i xi ≤ C
xi =∑
k λikLik + µikUik, ∀ i∑k λik + µik = 1, ∀ i
0 ≤ λik ≤ yik, ∀ i, k
0 ≤ µik ≤ yik, ∀ i, k∑k yik = 1, ∀ i
f(xi)
xiLik Uik
cik
dik
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Example 1: Production Planning: MILP
xi = quantity of product i (continuous)
yik = whether or not product i is made in mode k (binary)
λik, µik = weights for mode k (convex combination)
MILP Model
max∑
ik λikcik + µikdik∑i xi ≤ C
xi =∑
k λikLik + µikUik, ∀ i∑k λik + µik = 1, ∀ i
0 ≤ λik ≤ yik, ∀ i, k
0 ≤ µik ≤ yik, ∀ i, k∑k yik = 1, ∀ i
f(xi)
xiLik Uik
cik
dik
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Example 1: Production Planning: MILP
xi = quantity of product i (continuous)
yik = whether or not product i is made in mode k (binary)
λik, µik = weights for mode k (convex combination)
MILP Model
max∑
ik λikcik + µikdik∑i xi ≤ C
xi =∑
k λikLik + µikUik, ∀ i∑k λik + µik = 1, ∀ i
0 ≤ λik ≤ yik, ∀ i, k
0 ≤ µik ≤ yik, ∀ i, k∑k yik = 1, ∀ i
f(xi)
xiLik Uik
cik
dik
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Example 1: Production Planning: MILP
xi = quantity of product i (continuous)
yik = whether or not product i is made in mode k (binary)
λik, µik = weights for mode k (convex combination)
MILP Model
max∑
ik λikcik + µikdik∑i xi ≤ C
xi =∑
k λikLik + µikUik, ∀ i∑k λik + µik = 1, ∀ i
0 ≤ λik ≤ yik, ∀ i, k
0 ≤ µik ≤ yik, ∀ i, k∑k yik = 1, ∀ i
f(xi)
xiLik Uik
cik
dik
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Example 1: Production Planning: MILP
xi = quantity of product i (continuous)
yik = whether or not product i is made in mode k (binary)
λik, µik = weights for mode k (convex combination)
MILP Model
max∑
ik λikcik + µikdik∑i xi ≤ C
xi =∑
k λikLik + µikUik, ∀ i∑k λik + µik = 1, ∀ i
0 ≤ λik ≤ yik, ∀ i, k
0 ≤ µik ≤ yik, ∀ i, k∑k yik = 1, ∀ i
f(xi)
xiLik Uik
cik
dik
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 16
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Example 1: Production Planning: MILP
xi = quantity of product i (continuous)
yik = whether or not product i is made in mode k (binary)
λik, µik = weights for mode k (convex combination)
MILP Model
max∑
ik λikcik + µikdik
∑i xi ≤ C
xi =∑
k λikLik + µikUik, ∀ i∑k λik + µik = 1, ∀ i
0 ≤ λik ≤ yik, ∀ i, k
0 ≤ µik ≤ yik, ∀ i, k∑k yik = 1, ∀ i
f(xi)
xiLik Uik
cik
dik
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Example 1: Production Planning: MILP
xi = quantity of product i (continuous)
yik = whether or not product i is made in mode k (binary)
λik, µik = weights for mode k (convex combination)
MILP Model
max∑
ik λikcik + µikdik∑i xi ≤ C
xi =∑
k λikLik + µikUik, ∀ i∑k λik + µik = 1, ∀ i
0 ≤ λik ≤ yik, ∀ i, k
0 ≤ µik ≤ yik, ∀ i, k∑k yik = 1, ∀ i
f(xi)
xiLik Uik
cik
dik
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 16
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Example 1: Production Planning: MILP
xi = quantity of product i (continuous)
yik = whether or not product i is made in mode k (binary)
λik, µik = weights for mode k (convex combination)
MILP Model
max∑
ik λikcik + µikdik∑i xi ≤ C
xi =∑
k λikLik + µikUik, ∀ i
∑k λik + µik = 1, ∀ i
0 ≤ λik ≤ yik, ∀ i, k
0 ≤ µik ≤ yik, ∀ i, k∑k yik = 1, ∀ i
f(xi)
xiLik Uik
cik
dik
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Example 1: Production Planning: MILP
xi = quantity of product i (continuous)
yik = whether or not product i is made in mode k (binary)
λik, µik = weights for mode k (convex combination)
MILP Model
max∑
ik λikcik + µikdik∑i xi ≤ C
xi =∑
k λikLik + µikUik, ∀ i∑k λik + µik = 1, ∀ i
0 ≤ λik ≤ yik, ∀ i, k
0 ≤ µik ≤ yik, ∀ i, k∑k yik = 1, ∀ i
f(xi)
xiLik Uik
cik
dik
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Example 1: Production Planning: MILP
xi = quantity of product i (continuous)
yik = whether or not product i is made in mode k (binary)
λik, µik = weights for mode k (convex combination)
MILP Model
max∑
ik λikcik + µikdik∑i xi ≤ C
xi =∑
k λikLik + µikUik, ∀ i∑k λik + µik = 1, ∀ i
0 ≤ λik ≤ yik, ∀ i, k
0 ≤ µik ≤ yik, ∀ i, k∑k yik = 1, ∀ i
f(xi)
xiLik Uik
cik
dik
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Example 1: Production Planning: MILP
xi = quantity of product i (continuous)
yik = whether or not product i is made in mode k (binary)
λik, µik = weights for mode k (convex combination)
MILP Model
max∑
ik λikcik + µikdik∑i xi ≤ C
xi =∑
k λikLik + µikUik, ∀ i∑k λik + µik = 1, ∀ i
0 ≤ λik ≤ yik, ∀ i, k
0 ≤ µik ≤ yik, ∀ i, k
∑k yik = 1, ∀ i
f(xi)
xiLik Uik
cik
dik
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Example 1: Production Planning: MILP
xi = quantity of product i (continuous)
yik = whether or not product i is made in mode k (binary)
λik, µik = weights for mode k (convex combination)
MILP Model
max∑
ik λikcik + µikdik∑i xi ≤ C
xi =∑
k λikLik + µikUik, ∀ i∑k λik + µik = 1, ∀ i
0 ≤ λik ≤ yik, ∀ i, k
0 ≤ µik ≤ yik, ∀ i, k∑k yik = 1, ∀ i
f(xi)
xiLik Uik
cik
dik
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 16
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Example 1: Production Planning: Integrated
xi = quantity of product i (continuous)
ui = net income from product i (continuous)
Integrated Model
max∑
i ui∑i xi ≤ C
piecewise(xi, ui, Li, Ui, ci, di), ∀ i
f(xi)
xiLik Uik
cik
dik
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Example 1: Production Planning: Integrated
xi = quantity of product i (continuous)
ui = net income from product i (continuous)
Integrated Model
max∑
i ui∑i xi ≤ C
piecewise(xi, ui, Li, Ui, ci, di), ∀ i
f(xi)
xiLik Uik
cik
dik
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 17
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Example 1: Production Planning: Integrated
xi = quantity of product i (continuous)
ui = net income from product i (continuous)
Integrated Model
max∑
i ui∑i xi ≤ C
piecewise(xi, ui, Li, Ui, ci, di), ∀ i
f(xi)
xiLik Uik
cik
dik
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 17
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Example 1: Production Planning: Integrated
xi = quantity of product i (continuous)
ui = net income from product i (continuous)
Integrated Model
max∑
i ui∑i xi ≤ C
piecewise(xi, ui, Li, Ui, ci, di), ∀ i
f(xi)
xiLik Uik
cik
dik
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 17
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Example 1: Production Planning: Integrated
xi = quantity of product i (continuous)
ui = net income from product i (continuous)
Integrated Model
max∑
i ui
∑i xi ≤ C
piecewise(xi, ui, Li, Ui, ci, di), ∀ i
f(xi)
xiLik Uik
cik
dik
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 17
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Example 1: Production Planning: Integrated
xi = quantity of product i (continuous)
ui = net income from product i (continuous)
Integrated Model
max∑
i ui∑i xi ≤ C
piecewise(xi, ui, Li, Ui, ci, di), ∀ i
f(xi)
xiLik Uik
cik
dik
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 17
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Example 1: Production Planning: Integrated
xi = quantity of product i (continuous)
ui = net income from product i (continuous)
Integrated Model
max∑
i ui∑i xi ≤ C
piecewise(xi, ui, Li, Ui, ci, di), ∀ i
f(xi)
xiLik Uik
cik
dik
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 17
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Example 1: Production Planning: SIMPL Model
xi = quantity of product i (continuous)
ui = net income from product i (continuous)
SIMPL Model
OBJECTIVEmax sum i of u[i]
CONSTRAINTScapacity means {
sum i of x[i] <= Crelaxation = { lp, cp } }
income means {piecewise(x[i],u[i],L[i],U[i],c[i],d[i]) forall irelaxation = { lp, cp } }
SEARCHtype = { bb:bestdive }branching = { income:most }
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Example 1: Production Planning: SIMPL Model
xi = quantity of product i (continuous)
ui = net income from product i (continuous)
SIMPL Model
OBJECTIVEmax sum i of u[i]
CONSTRAINTScapacity means {
sum i of x[i] <= Crelaxation = { lp, cp } }
income means {piecewise(x[i],u[i],L[i],U[i],c[i],d[i]) forall irelaxation = { lp, cp } }
SEARCHtype = { bb:bestdive }branching = { income:most }
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Example 1: Production Planning: SIMPL Model
xi = quantity of product i (continuous)
ui = net income from product i (continuous)
SIMPL Model
OBJECTIVEmax sum i of u[i]
CONSTRAINTScapacity means {
sum i of x[i] <= Crelaxation = { lp, cp } }
income means {piecewise(x[i],u[i],L[i],U[i],c[i],d[i]) forall irelaxation = { lp, cp } }
SEARCHtype = { bb:bestdive }branching = { income:most }
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Example 1: Production Planning: SIMPL Model
xi = quantity of product i (continuous)
ui = net income from product i (continuous)
SIMPL Model
OBJECTIVEmax sum i of u[i]
CONSTRAINTScapacity means {
sum i of x[i] <= Crelaxation = { lp, cp } }
income means {piecewise(x[i],u[i],L[i],U[i],c[i],d[i]) forall irelaxation = { lp, cp } }
SEARCHtype = { bb:bestdive }branching = { income:most }
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 18
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Example 1: Production Planning: SIMPL Model
xi = quantity of product i (continuous)
ui = net income from product i (continuous)
SIMPL ModelOBJECTIVE
max sum i of u[i]CONSTRAINTS
capacity means {sum i of x[i] <= Crelaxation = { lp, cp } }
income means {piecewise(x[i],u[i],L[i],U[i],c[i],d[i]) forall irelaxation = { lp, cp } }
SEARCHtype = { bb:bestdive }branching = { income:most }
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Example 1: Production Planning: SIMPL Model
xi = quantity of product i (continuous)
ui = net income from product i (continuous)
SIMPL ModelOBJECTIVE
max sum i of u[i]
CONSTRAINTScapacity means {
sum i of x[i] <= Crelaxation = { lp, cp } }
income means {piecewise(x[i],u[i],L[i],U[i],c[i],d[i]) forall irelaxation = { lp, cp } }
SEARCHtype = { bb:bestdive }branching = { income:most }
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 18
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Example 1: Production Planning: SIMPL Model
xi = quantity of product i (continuous)
ui = net income from product i (continuous)
SIMPL ModelOBJECTIVE
max sum i of u[i]CONSTRAINTS
capacity means {sum i of x[i] <= Crelaxation = { lp, cp } }
income means {piecewise(x[i],u[i],L[i],U[i],c[i],d[i]) forall irelaxation = { lp, cp } }
SEARCHtype = { bb:bestdive }branching = { income:most }
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 18
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Example 1: Production Planning: SIMPL Model
xi = quantity of product i (continuous)
ui = net income from product i (continuous)
SIMPL ModelOBJECTIVE
max sum i of u[i]CONSTRAINTS
capacity means {sum i of x[i] <= Crelaxation = { lp, cp } }
income means {piecewise(x[i],u[i],L[i],U[i],c[i],d[i]) forall irelaxation = { lp, cp } }
SEARCHtype = { bb:bestdive }branching = { income:most }
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 18
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Example 1: Production Planning: SIMPL Model
xi = quantity of product i (continuous)
ui = net income from product i (continuous)
SIMPL ModelOBJECTIVE
max sum i of u[i]CONSTRAINTS
capacity means {sum i of x[i] <= Crelaxation = { lp, cp } }
income means {piecewise(x[i],u[i],L[i],U[i],c[i],d[i]) forall irelaxation = { lp, cp } }
SEARCHtype = { bb:bestdive }branching = { income:most }
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 18
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Example 1: Production Planning: SIMPL Model
xi = quantity of product i (continuous)
ui = net income from product i (continuous)
SIMPL ModelOBJECTIVE
max sum i of u[i]CONSTRAINTS
capacity means {sum i of x[i] <= Crelaxation = { lp, cp } }
income means {piecewise(x[i],u[i],L[i],U[i],c[i],d[i]) forall irelaxation = { lp, cp } }
SEARCH
type = { bb:bestdive }branching = { income:most }
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Example 1: Production Planning: SIMPL Model
xi = quantity of product i (continuous)
ui = net income from product i (continuous)
SIMPL ModelOBJECTIVE
max sum i of u[i]CONSTRAINTS
capacity means {sum i of x[i] <= Crelaxation = { lp, cp } }
income means {piecewise(x[i],u[i],L[i],U[i],c[i],d[i]) forall irelaxation = { lp, cp } }
SEARCHtype = { bb:bestdive }
branching = { income:most }
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 18
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Example 1: Production Planning: SIMPL Model
xi = quantity of product i (continuous)
ui = net income from product i (continuous)
SIMPL ModelOBJECTIVE
max sum i of u[i]CONSTRAINTS
capacity means {sum i of x[i] <= Crelaxation = { lp, cp } }
income means {piecewise(x[i],u[i],L[i],U[i],c[i],d[i]) forall irelaxation = { lp, cp } }
SEARCHtype = { bb:bestdive }branching = { income:most }
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Example 1: Production PlanningRelaxation and Branching for piecewise
f(x)
x
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Example 1: Production PlanningRelaxation and Branching for piecewise
f(x)
x
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Example 1: Production PlanningRelaxation and Branching for piecewise
f(x)
x
x value OK, y value OK: no problem
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Example 1: Production PlanningRelaxation and Branching for piecewise
f(x)
x
x value not OK: split domain
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Example 1: Production PlanningRelaxation and Branching for piecewise
f(x)
x
child 1
child 2
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Example 1: Production PlanningRelaxation and Branching for piecewise
f(x)
x
x OK, y not OK: 3-way branch on x
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Example 1: Production PlanningRelaxation and Branching for piecewise
f(x)
x
child 1
child 2
child 3
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Example 1: Production PlanningComputational Results: Number of Search Nodes
10 20 30 40 50
Number of Products
1
10
100
1000
10000
100000
Node
s MILP Integrated
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Example 1: Production PlanningComputational Results: Number of Search Nodes
10 20 30 40 50
Number of Products
1
10
100
1000
10000
100000
Node
s MILP Integrated
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Example 1: Production PlanningComputational Results: CPU Time (s)
10 20 30 40 50
Number of Products
0
2
4
6
8
10
12
14
Tim
e (s
)
MILP Integrated
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Example 1: Production PlanningComputational Results: CPU Time (s)
10 20 30 40 50
Number of Products
0
2
4
6
8
10
12
14
Tim
e (s
)
MILP Integrated
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Example 2: Product Configuration
A product (e.g. computer) is made up of several components(e.g. memory, cpu, etc.)
Components come in different types
Type k of component i uses/produces aijk units of resource j
cj = unit cost of resource j
Lower and upper bounds on resource usage/production
Objective: minimize total cost
Previous work: Thorsteinsson and Ottosson (2001)
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Example 2: Product Configuration
A product (e.g. computer) is made up of several components(e.g. memory, cpu, etc.)
Components come in different types
Type k of component i uses/produces aijk units of resource j
cj = unit cost of resource j
Lower and upper bounds on resource usage/production
Objective: minimize total cost
Previous work: Thorsteinsson and Ottosson (2001)
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 22
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Example 2: Product Configuration
A product (e.g. computer) is made up of several components(e.g. memory, cpu, etc.)
Components come in different types
Type k of component i uses/produces aijk units of resource j
cj = unit cost of resource j
Lower and upper bounds on resource usage/production
Objective: minimize total cost
Previous work: Thorsteinsson and Ottosson (2001)
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 22
![Page 106: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/106.jpg)
Example 2: Product Configuration
A product (e.g. computer) is made up of several components(e.g. memory, cpu, etc.)
Components come in different types
Type k of component i uses/produces aijk units of resource j
cj = unit cost of resource j
Lower and upper bounds on resource usage/production
Objective: minimize total cost
Previous work: Thorsteinsson and Ottosson (2001)
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 22
![Page 107: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/107.jpg)
Example 2: Product Configuration
A product (e.g. computer) is made up of several components(e.g. memory, cpu, etc.)
Components come in different types
Type k of component i uses/produces aijk units of resource j
cj = unit cost of resource j
Lower and upper bounds on resource usage/production
Objective: minimize total cost
Previous work: Thorsteinsson and Ottosson (2001)
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 22
![Page 108: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/108.jpg)
Example 2: Product Configuration
A product (e.g. computer) is made up of several components(e.g. memory, cpu, etc.)
Components come in different types
Type k of component i uses/produces aijk units of resource j
cj = unit cost of resource j
Lower and upper bounds on resource usage/production
Objective: minimize total cost
Previous work: Thorsteinsson and Ottosson (2001)
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 22
![Page 109: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/109.jpg)
Example 2: Product Configuration
A product (e.g. computer) is made up of several components(e.g. memory, cpu, etc.)
Components come in different types
Type k of component i uses/produces aijk units of resource j
cj = unit cost of resource j
Lower and upper bounds on resource usage/production
Objective: minimize total cost
Previous work: Thorsteinsson and Ottosson (2001)
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 22
![Page 110: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/110.jpg)
Example 2: Product Configuration
A product (e.g. computer) is made up of several components(e.g. memory, cpu, etc.)
Components come in different types
Type k of component i uses/produces aijk units of resource j
cj = unit cost of resource j
Lower and upper bounds on resource usage/production
Objective: minimize total cost
Previous work: Thorsteinsson and Ottosson (2001)
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 22
![Page 111: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/111.jpg)
Example 2: Product Configuration: MILP
xik = whether or not type k is chosen for component i (binary)
qik = # units of component i of type k to install (integer)
rj = amount of resource j produced (continuous)
MILP Model
min∑
j cjrj
rj =∑
ik aijkqik, ∀ j
Lj ≤ rj ≤ Uj, ∀ j
qik ≤ Mixik, ∀ i, k∑k xik = 1, ∀ i
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 23
![Page 112: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/112.jpg)
Example 2: Product Configuration: MILP
xik = whether or not type k is chosen for component i (binary)
qik = # units of component i of type k to install (integer)
rj = amount of resource j produced (continuous)
MILP Model
min∑
j cjrj
rj =∑
ik aijkqik, ∀ j
Lj ≤ rj ≤ Uj, ∀ j
qik ≤ Mixik, ∀ i, k∑k xik = 1, ∀ i
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 23
![Page 113: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/113.jpg)
Example 2: Product Configuration: MILP
xik = whether or not type k is chosen for component i (binary)
qik = # units of component i of type k to install (integer)
rj = amount of resource j produced (continuous)
MILP Model
min∑
j cjrj
rj =∑
ik aijkqik, ∀ j
Lj ≤ rj ≤ Uj, ∀ j
qik ≤ Mixik, ∀ i, k∑k xik = 1, ∀ i
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 23
![Page 114: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/114.jpg)
Example 2: Product Configuration: MILP
xik = whether or not type k is chosen for component i (binary)
qik = # units of component i of type k to install (integer)
rj = amount of resource j produced (continuous)
MILP Model
min∑
j cjrj
rj =∑
ik aijkqik, ∀ j
Lj ≤ rj ≤ Uj, ∀ j
qik ≤ Mixik, ∀ i, k∑k xik = 1, ∀ i
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 23
![Page 115: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/115.jpg)
Example 2: Product Configuration: MILP
xik = whether or not type k is chosen for component i (binary)
qik = # units of component i of type k to install (integer)
rj = amount of resource j produced (continuous)
MILP Model
min∑
j cjrj
rj =∑
ik aijkqik, ∀ j
Lj ≤ rj ≤ Uj, ∀ j
qik ≤ Mixik, ∀ i, k∑k xik = 1, ∀ i
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 23
![Page 116: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/116.jpg)
Example 2: Product Configuration: MILP
xik = whether or not type k is chosen for component i (binary)
qik = # units of component i of type k to install (integer)
rj = amount of resource j produced (continuous)
MILP Model
min∑
j cjrj
rj =∑
ik aijkqik, ∀ j
Lj ≤ rj ≤ Uj, ∀ j
qik ≤ Mixik, ∀ i, k∑k xik = 1, ∀ i
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 23
![Page 117: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/117.jpg)
Example 2: Product Configuration: MILP
xik = whether or not type k is chosen for component i (binary)
qik = # units of component i of type k to install (integer)
rj = amount of resource j produced (continuous)
MILP Model
min∑
j cjrj
rj =∑
ik aijkqik, ∀ j
Lj ≤ rj ≤ Uj, ∀ j
qik ≤ Mixik, ∀ i, k∑k xik = 1, ∀ i
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 23
![Page 118: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/118.jpg)
Example 2: Product Configuration: MILP
xik = whether or not type k is chosen for component i (binary)
qik = # units of component i of type k to install (integer)
rj = amount of resource j produced (continuous)
MILP Model
min∑
j cjrj
rj =∑
ik aijkqik, ∀ j
Lj ≤ rj ≤ Uj, ∀ j
qik ≤ Mixik, ∀ i, k∑k xik = 1, ∀ i
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 23
![Page 119: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/119.jpg)
Example 2: Product Configuration: MILP
xik = whether or not type k is chosen for component i (binary)
qik = # units of component i of type k to install (integer)
rj = amount of resource j produced (continuous)
MILP Model
min∑
j cjrj
rj =∑
ik aijkqik, ∀ j
Lj ≤ rj ≤ Uj, ∀ j
qik ≤ Mixik, ∀ i, k
∑k xik = 1, ∀ i
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 23
![Page 120: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/120.jpg)
Example 2: Product Configuration: MILP
xik = whether or not type k is chosen for component i (binary)
qik = # units of component i of type k to install (integer)
rj = amount of resource j produced (continuous)
MILP Model
min∑
j cjrj
rj =∑
ik aijkqik, ∀ j
Lj ≤ rj ≤ Uj, ∀ j
qik ≤ Mixik, ∀ i, k∑k xik = 1, ∀ i
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 23
![Page 121: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/121.jpg)
Example 2: Product Configuration: Integrated
qi = # units of component i to install (integer)ti = type chosen for component i (discrete)rj = amount of resource j produced (continuous)
Integrated Model
min∑
j cjrj
rj =∑
i aijtiqi, ∀ j
Lj ≤ rj ≤ Uj, ∀ j
zij = aijtiqi and element(ti, (aij1qi, . . . , aijnqi), zij)
∨k∈Dti
(zij = aijkqi) → automatic and dynamic convex hull relax.
converted to
equiv
alent
to
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 24
![Page 122: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/122.jpg)
Example 2: Product Configuration: Integrated
qi = # units of component i to install (integer)
ti = type chosen for component i (discrete)rj = amount of resource j produced (continuous)
Integrated Model
min∑
j cjrj
rj =∑
i aijtiqi, ∀ j
Lj ≤ rj ≤ Uj, ∀ j
zij = aijtiqi and element(ti, (aij1qi, . . . , aijnqi), zij)
∨k∈Dti
(zij = aijkqi) → automatic and dynamic convex hull relax.
converted to
equiv
alent
to
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 24
![Page 123: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/123.jpg)
Example 2: Product Configuration: Integrated
qi = # units of component i to install (integer)ti = type chosen for component i (discrete)
rj = amount of resource j produced (continuous)
Integrated Model
min∑
j cjrj
rj =∑
i aijtiqi, ∀ j
Lj ≤ rj ≤ Uj, ∀ j
zij = aijtiqi and element(ti, (aij1qi, . . . , aijnqi), zij)
∨k∈Dti
(zij = aijkqi) → automatic and dynamic convex hull relax.
converted to
equiv
alent
to
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 24
![Page 124: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/124.jpg)
Example 2: Product Configuration: Integrated
qi = # units of component i to install (integer)ti = type chosen for component i (discrete)rj = amount of resource j produced (continuous)
Integrated Model
min∑
j cjrj
rj =∑
i aijtiqi, ∀ j
Lj ≤ rj ≤ Uj, ∀ j
zij = aijtiqi and element(ti, (aij1qi, . . . , aijnqi), zij)
∨k∈Dti
(zij = aijkqi) → automatic and dynamic convex hull relax.
converted to
equiv
alent
to
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 24
![Page 125: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/125.jpg)
Example 2: Product Configuration: Integrated
qi = # units of component i to install (integer)ti = type chosen for component i (discrete)rj = amount of resource j produced (continuous)
Integrated Model
min∑
j cjrj
rj =∑
i aijtiqi, ∀ j
Lj ≤ rj ≤ Uj, ∀ j
zij = aijtiqi and element(ti, (aij1qi, . . . , aijnqi), zij)
∨k∈Dti
(zij = aijkqi) → automatic and dynamic convex hull relax.
converted to
equiv
alent
to
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 24
![Page 126: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/126.jpg)
Example 2: Product Configuration: Integrated
qi = # units of component i to install (integer)ti = type chosen for component i (discrete)rj = amount of resource j produced (continuous)
Integrated Model
min∑
j cjrj
rj =∑
i aijtiqi, ∀ j
Lj ≤ rj ≤ Uj, ∀ j
zij = aijtiqi and element(ti, (aij1qi, . . . , aijnqi), zij)
∨k∈Dti
(zij = aijkqi) → automatic and dynamic convex hull relax.
converted to
equiv
alent
to
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 24
![Page 127: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/127.jpg)
Example 2: Product Configuration: Integrated
qi = # units of component i to install (integer)ti = type chosen for component i (discrete)rj = amount of resource j produced (continuous)
Integrated Model
min∑
j cjrj
rj =∑
i aijtiqi, ∀ j
Lj ≤ rj ≤ Uj, ∀ j
zij = aijtiqi and element(ti, (aij1qi, . . . , aijnqi), zij)
∨k∈Dti
(zij = aijkqi) → automatic and dynamic convex hull relax.
converted to
equiv
alent
to
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 24
![Page 128: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/128.jpg)
Example 2: Product Configuration: Integrated
qi = # units of component i to install (integer)ti = type chosen for component i (discrete)rj = amount of resource j produced (continuous)
Integrated Model
min∑
j cjrj
rj =∑
i aijtiqi, ∀ j
Lj ≤ rj ≤ Uj, ∀ j
zij = aijtiqi and element(ti, (aij1qi, . . . , aijnqi), zij)
∨k∈Dti
(zij = aijkqi) → automatic and dynamic convex hull relax.
converted to
equiv
alent
to
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 24
![Page 129: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/129.jpg)
Example 2: Product Configuration: Integrated
qi = # units of component i to install (integer)ti = type chosen for component i (discrete)rj = amount of resource j produced (continuous)
Integrated Model
min∑
j cjrj
rj =∑
i aijtiqi, ∀ j
Lj ≤ rj ≤ Uj, ∀ j
zij = aijtiqi
and element(ti, (aij1qi, . . . , aijnqi), zij)
∨k∈Dti
(zij = aijkqi) → automatic and dynamic convex hull relax.
converted to
equiv
alent
to
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 24
![Page 130: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/130.jpg)
Example 2: Product Configuration: Integrated
qi = # units of component i to install (integer)ti = type chosen for component i (discrete)rj = amount of resource j produced (continuous)
Integrated Model
min∑
j cjrj
rj =∑
i aijtiqi, ∀ j
Lj ≤ rj ≤ Uj, ∀ j
zij = aijtiqi and element(ti, (aij1qi, . . . , aijnqi), zij)
∨k∈Dti
(zij = aijkqi) → automatic and dynamic convex hull relax.
converted to
equiv
alent
to
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 24
![Page 131: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/131.jpg)
Example 2: Product Configuration: Integrated
qi = # units of component i to install (integer)ti = type chosen for component i (discrete)rj = amount of resource j produced (continuous)
Integrated Model
min∑
j cjrj
rj =∑
i aijtiqi, ∀ j
Lj ≤ rj ≤ Uj, ∀ j
zij = aijtiqi and element(ti, (aij1qi, . . . , aijnqi), zij)
∨k∈Dti
(zij = aijkqi)
→ automatic and dynamic convex hull relax.
converted to
equiv
alent
to
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 24
![Page 132: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/132.jpg)
Example 2: Product Configuration: Integrated
qi = # units of component i to install (integer)ti = type chosen for component i (discrete)rj = amount of resource j produced (continuous)
Integrated Model
min∑
j cjrj
rj =∑
i aijtiqi, ∀ j
Lj ≤ rj ≤ Uj, ∀ j
zij = aijtiqi and element(ti, (aij1qi, . . . , aijnqi), zij)
∨k∈Dti
(zij = aijkqi) → automatic and dynamic convex hull relax.
converted to
equiv
alent
to
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 24
![Page 133: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/133.jpg)
Example 2: Product Configuration: SIMPL Model
OBJECTIVE min sum j of c[j]*r[j]CONSTRAINTS
resource means {r[j] = sum i of a[i][j][t[i]]*q[i] forall jrelaxation = { lp, cp } }
quant means {q[1] >= 1 => q[2] = 0relaxation = { lp, cp } }
types means {t[1] = 1 => t[2] in {1, 2}t[3] = 1 => (t[4] in {1, 3} and t[5] in {1, 3, 4, 6}
and t[6] = 3)relaxation = { lp, cp } }
SEARCHtype = { bb:bestdive }branching = { quant, t:most, q:least:triple, types:most }inference = { q:redcost }
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 25
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Example 2: Product Configuration: SIMPL Model
OBJECTIVE min sum j of c[j]*r[j]
CONSTRAINTSresource means {
r[j] = sum i of a[i][j][t[i]]*q[i] forall jrelaxation = { lp, cp } }
quant means {q[1] >= 1 => q[2] = 0relaxation = { lp, cp } }
types means {t[1] = 1 => t[2] in {1, 2}t[3] = 1 => (t[4] in {1, 3} and t[5] in {1, 3, 4, 6}
and t[6] = 3)relaxation = { lp, cp } }
SEARCHtype = { bb:bestdive }branching = { quant, t:most, q:least:triple, types:most }inference = { q:redcost }
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 25
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Example 2: Product Configuration: SIMPL Model
OBJECTIVE min sum j of c[j]*r[j]CONSTRAINTS
resource means {r[j] = sum i of a[i][j][t[i]]*q[i] forall jrelaxation = { lp, cp } }
quant means {q[1] >= 1 => q[2] = 0relaxation = { lp, cp } }
types means {t[1] = 1 => t[2] in {1, 2}t[3] = 1 => (t[4] in {1, 3} and t[5] in {1, 3, 4, 6}
and t[6] = 3)relaxation = { lp, cp } }
SEARCHtype = { bb:bestdive }branching = { quant, t:most, q:least:triple, types:most }inference = { q:redcost }
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 25
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Example 2: Product Configuration: SIMPL Model
OBJECTIVE min sum j of c[j]*r[j]CONSTRAINTS
resource means {r[j] = sum i of a[i][j][t[i]]*q[i] forall jrelaxation = { lp, cp } }
quant means {q[1] >= 1 => q[2] = 0relaxation = { lp, cp } }
types means {t[1] = 1 => t[2] in {1, 2}t[3] = 1 => (t[4] in {1, 3} and t[5] in {1, 3, 4, 6}
and t[6] = 3)relaxation = { lp, cp } }
SEARCHtype = { bb:bestdive }branching = { quant, t:most, q:least:triple, types:most }inference = { q:redcost }
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 25
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Example 2: Product Configuration: SIMPL Model
OBJECTIVE min sum j of c[j]*r[j]CONSTRAINTS
resource means {r[j] = sum i of a[i][j][t[i]]*q[i] forall jrelaxation = { lp, cp } }
quant means {q[1] >= 1 => q[2] = 0relaxation = { lp, cp } }
types means {t[1] = 1 => t[2] in {1, 2}t[3] = 1 => (t[4] in {1, 3} and t[5] in {1, 3, 4, 6}
and t[6] = 3)relaxation = { lp, cp } }
SEARCH
type = { bb:bestdive }branching = { quant, t:most, q:least:triple, types:most }inference = { q:redcost }
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 25
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Example 2: Product Configuration: SIMPL Model
OBJECTIVE min sum j of c[j]*r[j]CONSTRAINTS
resource means {r[j] = sum i of a[i][j][t[i]]*q[i] forall jrelaxation = { lp, cp } }
quant means {q[1] >= 1 => q[2] = 0relaxation = { lp, cp } }
types means {t[1] = 1 => t[2] in {1, 2}t[3] = 1 => (t[4] in {1, 3} and t[5] in {1, 3, 4, 6}
and t[6] = 3)relaxation = { lp, cp } }
SEARCHtype = { bb:bestdive }
branching = { quant, t:most, q:least:triple, types:most }inference = { q:redcost }
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 25
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Example 2: Product Configuration: SIMPL Model
OBJECTIVE min sum j of c[j]*r[j]CONSTRAINTS
resource means {r[j] = sum i of a[i][j][t[i]]*q[i] forall jrelaxation = { lp, cp } }
quant means {q[1] >= 1 => q[2] = 0relaxation = { lp, cp } }
types means {t[1] = 1 => t[2] in {1, 2}t[3] = 1 => (t[4] in {1, 3} and t[5] in {1, 3, 4, 6}
and t[6] = 3)relaxation = { lp, cp } }
SEARCHtype = { bb:bestdive }branching = {
quant,
t:most, q:least:triple
, types:most
}
inference = { q:redcost }
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 25
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Example 2: Product Configuration: SIMPL Model
OBJECTIVE min sum j of c[j]*r[j]CONSTRAINTS
resource means {r[j] = sum i of a[i][j][t[i]]*q[i] forall jrelaxation = { lp, cp } }
quant means {q[1] >= 1 => q[2] = 0relaxation = { lp, cp } }
types means {t[1] = 1 => t[2] in {1, 2}t[3] = 1 => (t[4] in {1, 3} and t[5] in {1, 3, 4, 6}
and t[6] = 3)relaxation = { lp, cp } }
SEARCHtype = { bb:bestdive }branching = {
quant,
t:most, q:least:triple
, types:most
}inference = { q:redcost }
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 25
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Example 2: Product Configuration: SIMPL Model
OBJECTIVE min sum j of c[j]*r[j]CONSTRAINTS
resource means {r[j] = sum i of a[i][j][t[i]]*q[i] forall jrelaxation = { lp, cp } }
quant means {q[1] >= 1 => q[2] = 0relaxation = { lp, cp } }
types means {t[1] = 1 => t[2] in {1, 2}t[3] = 1 => (t[4] in {1, 3} and t[5] in {1, 3, 4, 6}
and t[6] = 3)relaxation = { lp, cp } }
SEARCHtype = { bb:bestdive }branching = {
quant,
t:most, q:least:triple
, types:most
}inference = { q:redcost }
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 25
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Example 2: Product Configuration: SIMPL Model
OBJECTIVE min sum j of c[j]*r[j]CONSTRAINTS
resource means {r[j] = sum i of a[i][j][t[i]]*q[i] forall jrelaxation = { lp, cp } }
quant means {q[1] >= 1 => q[2] = 0relaxation = { lp, cp } }
types means {t[1] = 1 => t[2] in {1, 2}t[3] = 1 => (t[4] in {1, 3} and t[5] in {1, 3, 4, 6}
and t[6] = 3)relaxation = { lp, cp } }
SEARCHtype = { bb:bestdive }branching = {
quant,
t:most, q:least:triple
, types:most
}inference = { q:redcost }
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 25
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Example 2: Product Configuration: SIMPL Model
OBJECTIVE min sum j of c[j]*r[j]CONSTRAINTS
resource means {r[j] = sum i of a[i][j][t[i]]*q[i] forall jrelaxation = { lp, cp } }
quant means {q[1] >= 1 => q[2] = 0relaxation = { lp, cp } }
types means {t[1] = 1 => t[2] in {1, 2}t[3] = 1 => (t[4] in {1, 3} and t[5] in {1, 3, 4, 6}
and t[6] = 3)relaxation = { lp, cp } }
SEARCHtype = { bb:bestdive }branching = { quant, t:most, q:least:triple
, types:most
}inference = { q:redcost }
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 25
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Example 2: Product Configuration: SIMPL Model
OBJECTIVE min sum j of c[j]*r[j]CONSTRAINTS
resource means {r[j] = sum i of a[i][j][t[i]]*q[i] forall jrelaxation = { lp, cp } }
quant means {q[1] >= 1 => q[2] = 0relaxation = { lp, cp } }
types means {t[1] = 1 => t[2] in {1, 2}t[3] = 1 => (t[4] in {1, 3} and t[5] in {1, 3, 4, 6}
and t[6] = 3)relaxation = { lp, cp } }
SEARCHtype = { bb:bestdive }branching = { quant, t:most, q:least:triple, types:most }inference = { q:redcost }
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 25
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Example 2: Product ConfigurationComputational Results: Number of Search Nodes
2 4 6 8 10
Problem Instances (Thorsteinsson and Ottosson, 2001)
0
100
200
300
400
500
600
700
Node
s
MILP Integrated
1 3 5 7 9
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Example 2: Product ConfigurationComputational Results: Number of Search Nodes
2 4 6 8 10
Problem Instances (Thorsteinsson and Ottosson, 2001)
0
100
200
300
400
500
600
700
Node
s
MILP Integrated
1 3 5 7 9
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 26
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Example 2: Product ConfigurationComputational Results: CPU Time (s)
2 4 6 8 10
Problem Instances (Thorsteinsson and Ottosson, 2001)
0
5
10
15
20
Tim
e (s
)
MILP Integrated
1 3 5 7 9
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 27
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Example 2: Product ConfigurationComputational Results: CPU Time (s)
2 4 6 8 10
Problem Instances (Thorsteinsson and Ottosson, 2001)
0
5
10
15
20
Tim
e (s
)
MILP Integrated
1 3 5 7 9
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 27
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Example 3: Job Scheduling on Parallel Machines
Given n jobs and m parallel (disjunctive) machines
cij and pij = processing cost and time of job i on machine j
Job i has release date ri and due date di
Objective: schedule all jobs and minimize total cost
Jain and Grossmann (2001)
I Hybrid MILP/CP Benders decomposition approachI Required development of special purpose codeI Up to 1000 times faster than commercial solvers
In SIMPL we can get the same results with very little effort
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 28
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Example 3: Job Scheduling on Parallel Machines
Given n jobs and m parallel (disjunctive) machines
cij and pij = processing cost and time of job i on machine j
Job i has release date ri and due date di
Objective: schedule all jobs and minimize total cost
Jain and Grossmann (2001)
I Hybrid MILP/CP Benders decomposition approachI Required development of special purpose codeI Up to 1000 times faster than commercial solvers
In SIMPL we can get the same results with very little effort
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 28
![Page 151: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/151.jpg)
Example 3: Job Scheduling on Parallel Machines
Given n jobs and m parallel (disjunctive) machines
cij and pij = processing cost and time of job i on machine j
Job i has release date ri and due date di
Objective: schedule all jobs and minimize total cost
Jain and Grossmann (2001)
I Hybrid MILP/CP Benders decomposition approachI Required development of special purpose codeI Up to 1000 times faster than commercial solvers
In SIMPL we can get the same results with very little effort
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 28
![Page 152: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/152.jpg)
Example 3: Job Scheduling on Parallel Machines
Given n jobs and m parallel (disjunctive) machines
cij and pij = processing cost and time of job i on machine j
Job i has release date ri and due date di
Objective: schedule all jobs and minimize total cost
Jain and Grossmann (2001)
I Hybrid MILP/CP Benders decomposition approachI Required development of special purpose codeI Up to 1000 times faster than commercial solvers
In SIMPL we can get the same results with very little effort
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 28
![Page 153: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/153.jpg)
Example 3: Job Scheduling on Parallel Machines
Given n jobs and m parallel (disjunctive) machines
cij and pij = processing cost and time of job i on machine j
Job i has release date ri and due date di
Objective: schedule all jobs and minimize total cost
Jain and Grossmann (2001)
I Hybrid MILP/CP Benders decomposition approachI Required development of special purpose codeI Up to 1000 times faster than commercial solvers
In SIMPL we can get the same results with very little effort
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 28
![Page 154: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/154.jpg)
Example 3: Job Scheduling on Parallel Machines
Given n jobs and m parallel (disjunctive) machines
cij and pij = processing cost and time of job i on machine j
Job i has release date ri and due date di
Objective: schedule all jobs and minimize total cost
Jain and Grossmann (2001)
I Hybrid MILP/CP Benders decomposition approachI Required development of special purpose codeI Up to 1000 times faster than commercial solvers
In SIMPL we can get the same results with very little effort
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 28
![Page 155: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/155.jpg)
Example 3: Job Scheduling on Parallel Machines
Given n jobs and m parallel (disjunctive) machines
cij and pij = processing cost and time of job i on machine j
Job i has release date ri and due date di
Objective: schedule all jobs and minimize total cost
Jain and Grossmann (2001)I Hybrid MILP/CP Benders decomposition approachI Required development of special purpose codeI Up to 1000 times faster than commercial solvers
In SIMPL we can get the same results with very little effort
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 28
![Page 156: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/156.jpg)
Example 3: Job Scheduling on Parallel Machines
Given n jobs and m parallel (disjunctive) machines
cij and pij = processing cost and time of job i on machine j
Job i has release date ri and due date di
Objective: schedule all jobs and minimize total cost
Jain and Grossmann (2001)I Hybrid MILP/CP Benders decomposition approachI Required development of special purpose codeI Up to 1000 times faster than commercial solvers
In SIMPL we can get the same results with very little effort
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 28
![Page 157: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/157.jpg)
Example 3: Job Scheduling: MILP Model
xij = whether or not job j is assigned to machine i (binary)yjk = whether or not j precedes k on some machine (binary)tj = start time of job j (continuous)
min∑
ij cijxij
rj ≤ tj ≤ dj −∑
i pijxij, ∀ j∑i xij = 1, ∀ j
yjk + ykj ≤ 1, ∀ k > j
yjk + ykj ≥ xij + xik − 1, ∀ k > j, i
yjk + ykj + xij + xi′k ≤ 2, ∀ k > j, i′ 6= i
tk ≥ tj +∑
i pijxij −M(1− yjk), ∀ k 6= j∑j pijxij ≤ maxj{dj} −minj{rj}, ∀ i
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 29
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Example 3: Job Scheduling: MILP Model
xij = whether or not job j is assigned to machine i (binary)
yjk = whether or not j precedes k on some machine (binary)tj = start time of job j (continuous)
min∑
ij cijxij
rj ≤ tj ≤ dj −∑
i pijxij, ∀ j∑i xij = 1, ∀ j
yjk + ykj ≤ 1, ∀ k > j
yjk + ykj ≥ xij + xik − 1, ∀ k > j, i
yjk + ykj + xij + xi′k ≤ 2, ∀ k > j, i′ 6= i
tk ≥ tj +∑
i pijxij −M(1− yjk), ∀ k 6= j∑j pijxij ≤ maxj{dj} −minj{rj}, ∀ i
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 29
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Example 3: Job Scheduling: MILP Model
xij = whether or not job j is assigned to machine i (binary)yjk = whether or not j precedes k on some machine (binary)
tj = start time of job j (continuous)
min∑
ij cijxij
rj ≤ tj ≤ dj −∑
i pijxij, ∀ j∑i xij = 1, ∀ j
yjk + ykj ≤ 1, ∀ k > j
yjk + ykj ≥ xij + xik − 1, ∀ k > j, i
yjk + ykj + xij + xi′k ≤ 2, ∀ k > j, i′ 6= i
tk ≥ tj +∑
i pijxij −M(1− yjk), ∀ k 6= j∑j pijxij ≤ maxj{dj} −minj{rj}, ∀ i
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 29
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Example 3: Job Scheduling: MILP Model
xij = whether or not job j is assigned to machine i (binary)yjk = whether or not j precedes k on some machine (binary)tj = start time of job j (continuous)
min∑
ij cijxij
rj ≤ tj ≤ dj −∑
i pijxij, ∀ j∑i xij = 1, ∀ j
yjk + ykj ≤ 1, ∀ k > j
yjk + ykj ≥ xij + xik − 1, ∀ k > j, i
yjk + ykj + xij + xi′k ≤ 2, ∀ k > j, i′ 6= i
tk ≥ tj +∑
i pijxij −M(1− yjk), ∀ k 6= j∑j pijxij ≤ maxj{dj} −minj{rj}, ∀ i
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 29
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Example 3: Job Scheduling: MILP Model
xij = whether or not job j is assigned to machine i (binary)yjk = whether or not j precedes k on some machine (binary)tj = start time of job j (continuous)
min∑
ij cijxij
rj ≤ tj ≤ dj −∑
i pijxij, ∀ j∑i xij = 1, ∀ j
yjk + ykj ≤ 1, ∀ k > j
yjk + ykj ≥ xij + xik − 1, ∀ k > j, i
yjk + ykj + xij + xi′k ≤ 2, ∀ k > j, i′ 6= i
tk ≥ tj +∑
i pijxij −M(1− yjk), ∀ k 6= j∑j pijxij ≤ maxj{dj} −minj{rj}, ∀ i
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Example 3: Job Scheduling on Parallel MachinesBenders Decomposition Approach
Master ProblemI Assign jobs to machines at minimum costI “Ignore” release dates and due datesI xij = 1 if job i assigned to machine j
Subproblem for machine jI Try to find feasible schedule with given set of jobs Ij
I If infeasible, generate Benders cut∑i∈Ij
xij ≤ |Ij| − 1
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Example 3: Job Scheduling on Parallel MachinesBenders Decomposition Approach
Master ProblemI Assign jobs to machines at minimum costI “Ignore” release dates and due datesI xij = 1 if job i assigned to machine j
Subproblem for machine jI Try to find feasible schedule with given set of jobs Ij
I If infeasible, generate Benders cut∑i∈Ij
xij ≤ |Ij| − 1
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 30
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Example 3: Job Scheduling on Parallel MachinesBenders Decomposition Approach
Master ProblemI Assign jobs to machines at minimum costI “Ignore” release dates and due datesI xij = 1 if job i assigned to machine j
Subproblem for machine jI Try to find feasible schedule with given set of jobs Ij
I If infeasible, generate Benders cut
∑i∈Ij
xij ≤ |Ij| − 1
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 30
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Example 3: Job Scheduling on Parallel MachinesBenders Decomposition Approach
Master ProblemI Assign jobs to machines at minimum costI “Ignore” release dates and due datesI xij = 1 if job i assigned to machine j
Subproblem for machine jI Try to find feasible schedule with given set of jobs Ij
I If infeasible, generate Benders cut∑i∈Ij
xij ≤ |Ij| − 1
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 30
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Example 3: Job Scheduling: Integrated Benders
xij = whether or not job j is assigned to machine i (binary)
yj = machine assigned to job j (integer)
tj = start time of job j (continuous)
Integrated Benders Model
min∑
ij cijxij∑i xij = 1, ∀ j
(xij = 1) ⇔ (yj = i), ∀ i, j
rj ≤ tj ≤ dj − pyjj, ∀ j
cumulative((tj, pij, 1 |xij = 1), 1), ∀ i
Need to tell the solver how to decompose the model
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 31
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Example 3: Job Scheduling: Integrated Benders
xij = whether or not job j is assigned to machine i (binary)
yj = machine assigned to job j (integer)
tj = start time of job j (continuous)
Integrated Benders Model
min∑
ij cijxij∑i xij = 1, ∀ j
(xij = 1) ⇔ (yj = i), ∀ i, j
rj ≤ tj ≤ dj − pyjj, ∀ j
cumulative((tj, pij, 1 |xij = 1), 1), ∀ i
Need to tell the solver how to decompose the model
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 31
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Example 3: Job Scheduling: Integrated Benders
xij = whether or not job j is assigned to machine i (binary)
yj = machine assigned to job j (integer)
tj = start time of job j (continuous)
Integrated Benders Model
min∑
ij cijxij∑i xij = 1, ∀ j
(xij = 1) ⇔ (yj = i), ∀ i, j
rj ≤ tj ≤ dj − pyjj, ∀ j
cumulative((tj, pij, 1 |xij = 1), 1), ∀ i
Need to tell the solver how to decompose the model
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 31
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Example 3: Job Scheduling: Integrated Benders
xij = whether or not job j is assigned to machine i (binary)
yj = machine assigned to job j (integer)
tj = start time of job j (continuous)
Integrated Benders Model
min∑
ij cijxij∑i xij = 1, ∀ j
(xij = 1) ⇔ (yj = i), ∀ i, j
rj ≤ tj ≤ dj − pyjj, ∀ j
cumulative((tj, pij, 1 |xij = 1), 1), ∀ i
Need to tell the solver how to decompose the model
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 31
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Example 3: Job Scheduling: Integrated Benders
xij = whether or not job j is assigned to machine i (binary)
yj = machine assigned to job j (integer)
tj = start time of job j (continuous)
Integrated Benders Model
min∑
ij cijxij∑i xij = 1, ∀ j
(xij = 1) ⇔ (yj = i), ∀ i, j
rj ≤ tj ≤ dj − pyjj, ∀ j
cumulative((tj, pij, 1 |xij = 1), 1), ∀ i
Need to tell the solver how to decompose the model
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 31
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Example 3: Job Scheduling: Integrated Benders
xij = whether or not job j is assigned to machine i (binary)
yj = machine assigned to job j (integer)
tj = start time of job j (continuous)
Integrated Benders Model
min∑
ij cijxij
∑i xij = 1, ∀ j
(xij = 1) ⇔ (yj = i), ∀ i, j
rj ≤ tj ≤ dj − pyjj, ∀ j
cumulative((tj, pij, 1 |xij = 1), 1), ∀ i
Need to tell the solver how to decompose the model
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 31
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Example 3: Job Scheduling: Integrated Benders
xij = whether or not job j is assigned to machine i (binary)
yj = machine assigned to job j (integer)
tj = start time of job j (continuous)
Integrated Benders Model
min∑
ij cijxij∑i xij = 1, ∀ j
(xij = 1) ⇔ (yj = i), ∀ i, j
rj ≤ tj ≤ dj − pyjj, ∀ j
cumulative((tj, pij, 1 |xij = 1), 1), ∀ i
Need to tell the solver how to decompose the model
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 31
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Example 3: Job Scheduling: Integrated Benders
xij = whether or not job j is assigned to machine i (binary)
yj = machine assigned to job j (integer)
tj = start time of job j (continuous)
Integrated Benders Model
min∑
ij cijxij∑i xij = 1, ∀ j
(xij = 1) ⇔ (yj = i), ∀ i, j
rj ≤ tj ≤ dj − pyjj, ∀ j
cumulative((tj, pij, 1 |xij = 1), 1), ∀ i
Need to tell the solver how to decompose the model
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 31
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Example 3: Job Scheduling: Integrated Benders
xij = whether or not job j is assigned to machine i (binary)
yj = machine assigned to job j (integer)
tj = start time of job j (continuous)
Integrated Benders Model
min∑
ij cijxij∑i xij = 1, ∀ j
(xij = 1) ⇔ (yj = i), ∀ i, j
rj ≤ tj ≤ dj − pyjj, ∀ j
cumulative((tj, pij, 1 |xij = 1), 1), ∀ i
Need to tell the solver how to decompose the model
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 31
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Example 3: Job Scheduling: Integrated Benders
xij = whether or not job j is assigned to machine i (binary)
yj = machine assigned to job j (integer)
tj = start time of job j (continuous)
Integrated Benders Model
min∑
ij cijxij∑i xij = 1, ∀ j
(xij = 1) ⇔ (yj = i), ∀ i, j
rj ≤ tj ≤ dj − pyjj, ∀ j
cumulative((tj, pij, 1 |xij = 1), 1), ∀ i
Need to tell the solver how to decompose the model
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 31
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Example 3: Job Scheduling: Integrated Benders
xij = whether or not job j is assigned to machine i (binary)
yj = machine assigned to job j (integer)
tj = start time of job j (continuous)
Integrated Benders Model
min∑
ij cijxij∑i xij = 1, ∀ j
(xij = 1) ⇔ (yj = i), ∀ i, j
rj ≤ tj ≤ dj − pyjj, ∀ j
cumulative((tj, pij, 1 |xij = 1), 1), ∀ i
Need to tell the solver how to decompose the model
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 31
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Example 3: Job Scheduling: SIMPL Model
OBJECTIVE min sum i,j of c[i][j]*x[i][j]CONSTRAINTSassign means {sum i of x[i][j] = 1 forall jrelaxation = { ip:master } }
xy means {x[i][j] = 1 <=> y[j] = i forall i, jrelaxation = { cp:sub } }
tbounds means {r[j] <= t[j] <= d[j] - p[y[j]][j] forall jrelaxation = { ip:master, cp:sub } }
machinecap means {cumulative({t[j],p[i][j],1} forall j | x[i][j]=1, 1) forall irelaxation = { ip:master, cp:sub:decomp } }inference = { feasibility } }
SEARCHtype = { benders }
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Example 3: Job Scheduling: SIMPL Model
OBJECTIVE min sum i,j of c[i][j]*x[i][j]
CONSTRAINTSassign means {
sum i of x[i][j] = 1 forall jrelaxation = { ip:master } }
xy means {x[i][j] = 1 <=> y[j] = i forall i, jrelaxation = { cp:sub } }
tbounds means {r[j] <= t[j] <= d[j] - p[y[j]][j] forall jrelaxation = { ip:master, cp:sub } }
machinecap means {cumulative({t[j],p[i][j],1} forall j | x[i][j]=1, 1) forall irelaxation = { ip:master, cp:sub:decomp } }inference = { feasibility } }
SEARCHtype = { benders }
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 32
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Example 3: Job Scheduling: SIMPL Model
OBJECTIVE min sum i,j of c[i][j]*x[i][j]CONSTRAINTS
assign means {sum i of x[i][j] = 1 forall jrelaxation = { ip:master } }
xy means {x[i][j] = 1 <=> y[j] = i forall i, jrelaxation = { cp:sub } }
tbounds means {r[j] <= t[j] <= d[j] - p[y[j]][j] forall jrelaxation = { ip:master, cp:sub } }
machinecap means {cumulative({t[j],p[i][j],1} forall j | x[i][j]=1, 1) forall irelaxation = { ip:master, cp:sub:decomp } }inference = { feasibility } }
SEARCHtype = { benders }
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 32
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Example 3: Job Scheduling: SIMPL Model
OBJECTIVE min sum i,j of c[i][j]*x[i][j]CONSTRAINTSassign means {sum i of x[i][j] = 1 forall jrelaxation = { ip:master } }
xy means {x[i][j] = 1 <=> y[j] = i forall i, jrelaxation = { cp:sub } }
tbounds means {r[j] <= t[j] <= d[j] - p[y[j]][j] forall jrelaxation = { ip:master, cp:sub } }
machinecap means {cumulative({t[j],p[i][j],1} forall j | x[i][j]=1, 1) forall irelaxation = { ip:master, cp:sub:decomp } }inference = { feasibility } }
SEARCHtype = { benders }
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 32
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Example 3: Job Scheduling: SIMPL Model
OBJECTIVE min sum i,j of c[i][j]*x[i][j]CONSTRAINTSassign means {sum i of x[i][j] = 1 forall jrelaxation = { ip:master } }
xy means {x[i][j] = 1 <=> y[j] = i forall i, jrelaxation = { cp:sub } }
tbounds means {r[j] <= t[j] <= d[j] - p[y[j]][j] forall jrelaxation = { ip:master, cp:sub } }
machinecap means {cumulative({t[j],p[i][j],1} forall j | x[i][j]=1, 1) forall irelaxation = { ip:master, cp:sub:decomp } }inference = { feasibility } }
SEARCHtype = { benders }
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 32
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Example 3: Job Scheduling: SIMPL Model
OBJECTIVE min sum i,j of c[i][j]*x[i][j]CONSTRAINTSassign means {sum i of x[i][j] = 1 forall jrelaxation = { ip:master } }
xy means {x[i][j] = 1 <=> y[j] = i forall i, jrelaxation = { cp:sub } }
tbounds means {r[j] <= t[j] <= d[j] - p[y[j]][j] forall jrelaxation = { ip:master, cp:sub } }
machinecap means {cumulative({t[j],p[i][j],1} forall j | x[i][j]=1, 1) forall irelaxation = { ip:master, cp:sub:decomp } }inference = { feasibility } }
SEARCHtype = { benders }
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 32
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Example 3: Job Scheduling: SIMPL Model
OBJECTIVE min sum i,j of c[i][j]*x[i][j]CONSTRAINTSassign means {sum i of x[i][j] = 1 forall jrelaxation = { ip:master } }
xy means {x[i][j] = 1 <=> y[j] = i forall i, jrelaxation = { cp:sub } }
tbounds means {r[j] <= t[j] <= d[j] - p[y[j]][j] forall jrelaxation = { ip:master, cp:sub } }
machinecap means {cumulative({t[j],p[i][j],1} forall j | x[i][j]=1, 1) forall irelaxation = { ip:master, cp:sub:decomp } }inference = { feasibility } }
SEARCHtype = { benders }
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 32
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Example 3: Job Scheduling: SIMPL Model
OBJECTIVE min sum i,j of c[i][j]*x[i][j]CONSTRAINTSassign means {sum i of x[i][j] = 1 forall jrelaxation = { ip:master } }
xy means {x[i][j] = 1 <=> y[j] = i forall i, jrelaxation = { cp:sub } }
tbounds means {r[j] <= t[j] <= d[j] - p[y[j]][j] forall jrelaxation = { ip:master, cp:sub } }
machinecap means {cumulative({t[j],p[i][j],1} forall j | x[i][j]=1, 1) forall irelaxation = { ip:master, cp:sub:decomp } }inference = { feasibility } }
SEARCHtype = { benders }
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 32
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Example 3: Job Scheduling on Parallel MachinesComputational Results
Instances from Jain and Grossmann (2001)
Long Processing Times
MILP Integrated BendersJobs Machines Nodes Time (s) Iterations Cuts Time (s)
3 2
1 0.00 2 1 0.00
7 3
1 0.02 12 14 0.09
12 3
11060 16.50 26 37 0.58
15 5
3674 14.30 22 31 0.96
20 5
159400 3123.34 30 52 3.21
22 5
> 5.0M > 48h 38 59 6.70
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Example 3: Job Scheduling on Parallel MachinesComputational Results
Instances from Jain and Grossmann (2001)
Long Processing Times
MILP Integrated BendersJobs Machines Nodes Time (s) Iterations Cuts Time (s)
3 2
1 0.00 2 1 0.00
7 3
1 0.02 12 14 0.09
12 3
11060 16.50 26 37 0.58
15 5
3674 14.30 22 31 0.96
20 5
159400 3123.34 30 52 3.21
22 5
> 5.0M > 48h 38 59 6.70
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 33
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Example 3: Job Scheduling on Parallel MachinesComputational Results
Instances from Jain and Grossmann (2001)
Long Processing Times
MILP Integrated BendersJobs Machines Nodes Time (s) Iterations Cuts Time (s)
3 2 1 0.00
2 1 0.00
7 3 1 0.02
12 14 0.09
12 3 11060 16.50
26 37 0.58
15 5 3674 14.30
22 31 0.96
20 5 159400 3123.34
30 52 3.21
22 5 > 5.0M > 48h
38 59 6.70
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 33
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Example 3: Job Scheduling on Parallel MachinesComputational Results
Instances from Jain and Grossmann (2001)
Long Processing Times
MILP Integrated BendersJobs Machines Nodes Time (s) Iterations Cuts Time (s)
3 2 1 0.00 2 1 0.007 3 1 0.02 12 14 0.0912 3 11060 16.50 26 37 0.5815 5 3674 14.30 22 31 0.9620 5 159400 3123.34 30 52 3.2122 5 > 5.0M > 48h 38 59 6.70
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 33
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Example 3: Job Scheduling on Parallel MachinesComputational Results (continued)
Instances from Jain and Grossmann (2001)Shorter processing times make the problem easier to solve
Short Processing Times
MILP Integrated BendersJobs Machines Nodes Time (s) Iterations Cuts Time (s)
3 2
1 0.00 1 0 0.00
7 3
1 0.01 1 0 0.01
12 3
4950 1.98 1 0 0.01
15 5
14000 19.80 1 0 0.03
20 5
140 5.73 3 3 0.12
22 5
> 16.9M > 48h 5 4 0.38
25 5
> 4.5M > 48h 16 22 0.86
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 34
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Example 3: Job Scheduling on Parallel MachinesComputational Results (continued)
Instances from Jain and Grossmann (2001)Shorter processing times make the problem easier to solve
Short Processing Times
MILP Integrated BendersJobs Machines Nodes Time (s) Iterations Cuts Time (s)
3 2
1 0.00 1 0 0.00
7 3
1 0.01 1 0 0.01
12 3
4950 1.98 1 0 0.01
15 5
14000 19.80 1 0 0.03
20 5
140 5.73 3 3 0.12
22 5
> 16.9M > 48h 5 4 0.38
25 5
> 4.5M > 48h 16 22 0.86
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 34
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Example 3: Job Scheduling on Parallel MachinesComputational Results (continued)
Instances from Jain and Grossmann (2001)Shorter processing times make the problem easier to solve
Short Processing Times
MILP Integrated BendersJobs Machines Nodes Time (s) Iterations Cuts Time (s)
3 2 1 0.00
1 0 0.00
7 3 1 0.01
1 0 0.01
12 3 4950 1.98
1 0 0.01
15 5 14000 19.80
1 0 0.03
20 5 140 5.73
3 3 0.12
22 5 > 16.9M > 48h
5 4 0.38
25 5 > 4.5M > 48h
16 22 0.86
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 34
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Example 3: Job Scheduling on Parallel MachinesComputational Results (continued)
Instances from Jain and Grossmann (2001)Shorter processing times make the problem easier to solve
Short Processing Times
MILP Integrated BendersJobs Machines Nodes Time (s) Iterations Cuts Time (s)
3 2 1 0.00 1 0 0.007 3 1 0.01 1 0 0.0112 3 4950 1.98 1 0 0.0115 5 14000 19.80 1 0 0.0320 5 140 5.73 3 3 0.1222 5 > 16.9M > 48h 5 4 0.3825 5 > 4.5M > 48h 16 22 0.86
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 34
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Future Work
Regarding SIMPL itself...
I Support other integrated approaches, e.g. local search, B&PI More features: non-linear solver, cutting planes, more
constraintsI More powerful language in SEARCH section (like OPL)I More intelligent model compilation (e.g. detect special
structures)I Improve performance (code optimization)I etc.
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 35
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Future Work
Regarding SIMPL itself...
I Support other integrated approaches, e.g. local search, B&PI More features: non-linear solver, cutting planes, more
constraintsI More powerful language in SEARCH section (like OPL)I More intelligent model compilation (e.g. detect special
structures)I Improve performance (code optimization)I etc.
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 35
![Page 195: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/195.jpg)
Future Work
Regarding SIMPL itself...I Support other integrated approaches, e.g. local search, B&P
I More features: non-linear solver, cutting planes, moreconstraints
I More powerful language in SEARCH section (like OPL)I More intelligent model compilation (e.g. detect special
structures)I Improve performance (code optimization)I etc.
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 35
![Page 196: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/196.jpg)
Future Work
Regarding SIMPL itself...I Support other integrated approaches, e.g. local search, B&PI More features: non-linear solver, cutting planes, more
constraints
I More powerful language in SEARCH section (like OPL)I More intelligent model compilation (e.g. detect special
structures)I Improve performance (code optimization)I etc.
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 35
![Page 197: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/197.jpg)
Future Work
Regarding SIMPL itself...I Support other integrated approaches, e.g. local search, B&PI More features: non-linear solver, cutting planes, more
constraintsI More powerful language in SEARCH section (like OPL)
I More intelligent model compilation (e.g. detect specialstructures)
I Improve performance (code optimization)I etc.
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 35
![Page 198: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/198.jpg)
Future Work
Regarding SIMPL itself...I Support other integrated approaches, e.g. local search, B&PI More features: non-linear solver, cutting planes, more
constraintsI More powerful language in SEARCH section (like OPL)I More intelligent model compilation (e.g. detect special
structures)
I Improve performance (code optimization)I etc.
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 35
![Page 199: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/199.jpg)
Future Work
Regarding SIMPL itself...I Support other integrated approaches, e.g. local search, B&PI More features: non-linear solver, cutting planes, more
constraintsI More powerful language in SEARCH section (like OPL)I More intelligent model compilation (e.g. detect special
structures)I Improve performance (code optimization)
I etc.
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 35
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Future Work
Regarding SIMPL itself...I Support other integrated approaches, e.g. local search, B&PI More features: non-linear solver, cutting planes, more
constraintsI More powerful language in SEARCH section (like OPL)I More intelligent model compilation (e.g. detect special
structures)I Improve performance (code optimization)I etc.
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 35
![Page 201: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/201.jpg)
Future Work (continued)
Regarding SIMPL’s availability...
I Distribute source code?I Distribute executable?I Add it to NEOS? COIN-OR?I We are still thinking about it...I Whichever way we go, we still need to clean it up a bit...
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 36
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Future Work (continued)
Regarding SIMPL’s availability...
I Distribute source code?I Distribute executable?I Add it to NEOS? COIN-OR?I We are still thinking about it...I Whichever way we go, we still need to clean it up a bit...
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 36
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Future Work (continued)
Regarding SIMPL’s availability...I Distribute source code?
I Distribute executable?I Add it to NEOS? COIN-OR?I We are still thinking about it...I Whichever way we go, we still need to clean it up a bit...
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 36
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Future Work (continued)
Regarding SIMPL’s availability...I Distribute source code?I Distribute executable?
I Add it to NEOS? COIN-OR?I We are still thinking about it...I Whichever way we go, we still need to clean it up a bit...
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 36
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Future Work (continued)
Regarding SIMPL’s availability...I Distribute source code?I Distribute executable?I Add it to NEOS? COIN-OR?
I We are still thinking about it...I Whichever way we go, we still need to clean it up a bit...
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 36
![Page 206: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/206.jpg)
Future Work (continued)
Regarding SIMPL’s availability...I Distribute source code?I Distribute executable?I Add it to NEOS? COIN-OR?I We are still thinking about it...
I Whichever way we go, we still need to clean it up a bit...
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 36
![Page 207: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/207.jpg)
Future Work (continued)
Regarding SIMPL’s availability...I Distribute source code?I Distribute executable?I Add it to NEOS? COIN-OR?I We are still thinking about it...I Whichever way we go, we still need to clean it up a bit...
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 36
![Page 208: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/208.jpg)
Conclusion
Many theoretical and technological breakthroughs over the last fewdecades have helped OR become more accessible and popular
Many important problems are still hard for traditional methods
Recent literature shows integrated methods can succeed whentraditional methods fail
SIMPL is
I a step toward making integrated methods more accessible to alarger group of users
I a very useful research tool
We still have a long way to go, but important steps have been takenand initial results are encouraging
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 37
![Page 209: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/209.jpg)
Conclusion
Many theoretical and technological breakthroughs over the last fewdecades have helped OR become more accessible and popular
Many important problems are still hard for traditional methods
Recent literature shows integrated methods can succeed whentraditional methods fail
SIMPL is
I a step toward making integrated methods more accessible to alarger group of users
I a very useful research tool
We still have a long way to go, but important steps have been takenand initial results are encouraging
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 37
![Page 210: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/210.jpg)
Conclusion
Many theoretical and technological breakthroughs over the last fewdecades have helped OR become more accessible and popular
Many important problems are still hard for traditional methods
Recent literature shows integrated methods can succeed whentraditional methods fail
SIMPL is
I a step toward making integrated methods more accessible to alarger group of users
I a very useful research tool
We still have a long way to go, but important steps have been takenand initial results are encouraging
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 37
![Page 211: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/211.jpg)
Conclusion
Many theoretical and technological breakthroughs over the last fewdecades have helped OR become more accessible and popular
Many important problems are still hard for traditional methods
Recent literature shows integrated methods can succeed whentraditional methods fail
SIMPL is
I a step toward making integrated methods more accessible to alarger group of users
I a very useful research tool
We still have a long way to go, but important steps have been takenand initial results are encouraging
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 37
![Page 212: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/212.jpg)
Conclusion
Many theoretical and technological breakthroughs over the last fewdecades have helped OR become more accessible and popular
Many important problems are still hard for traditional methods
Recent literature shows integrated methods can succeed whentraditional methods fail
SIMPL is
I a step toward making integrated methods more accessible to alarger group of users
I a very useful research tool
We still have a long way to go, but important steps have been takenand initial results are encouraging
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 37
![Page 213: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/213.jpg)
Conclusion
Many theoretical and technological breakthroughs over the last fewdecades have helped OR become more accessible and popular
Many important problems are still hard for traditional methods
Recent literature shows integrated methods can succeed whentraditional methods fail
SIMPL is
I a step toward making integrated methods more accessible to alarger group of users
I a very useful research tool
We still have a long way to go, but important steps have been takenand initial results are encouraging
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 37
![Page 214: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/214.jpg)
Conclusion
Many theoretical and technological breakthroughs over the last fewdecades have helped OR become more accessible and popular
Many important problems are still hard for traditional methods
Recent literature shows integrated methods can succeed whentraditional methods fail
SIMPL is
I a step toward making integrated methods more accessible to alarger group of users
I a very useful research tool
We still have a long way to go, but important steps have been takenand initial results are encouraging
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 37
![Page 215: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/215.jpg)
Conclusion
Many theoretical and technological breakthroughs over the last fewdecades have helped OR become more accessible and popular
Many important problems are still hard for traditional methods
Recent literature shows integrated methods can succeed whentraditional methods fail
SIMPL is
I a step toward making integrated methods more accessible to alarger group of users
I a very useful research tool
We still have a long way to go, but important steps have been takenand initial results are encouraging
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 37
![Page 216: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/216.jpg)
That’s All Folks!
Thank you!
Any Questions?
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 38
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Constraint Programming Example
x ∈ {1, 3}y ∈ {1, 2, 3}z ∈ {1, 2, 3}w ∈ {1, 2, 4}
element(z, [1, 3, 5], x) (x is the zth element of [1, 3, 5])
alldifferent(x, y, z, w) (all variables take distinct values)
2z − w ≥ 0
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 39
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Constraint Programming Example
x ∈ {1, 3}y ∈ {1, 2, 3}z ∈ {1, 2, 3}w ∈ {1, 2, 4}
element(z, [1, 3, 5], x) (x is the zth element of [1, 3, 5])
alldifferent(x, y, z, w) (all variables take distinct values)
2z − w ≥ 0
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 39
![Page 219: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/219.jpg)
Constraint Programming Example
x ∈ {1, 3}y ∈ {1, 2, 3}z ∈ {1, 2, 3}w ∈ {1, 2, 4}
element(z, [1, 3, 5], x) (x is the zth element of [1, 3, 5])
alldifferent(x, y, z, w) (all variables take distinct values)
2z − w ≥ 0
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 39
![Page 220: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/220.jpg)
Constraint Programming Example
x ∈ {1, 3}y ∈ {1, 2, 3}z ∈ {1, 2, 3}w ∈ {1, 2, 4}
element(z, [1, 3, 5], x) (x is the zth element of [1, 3, 5])
alldifferent(x, y, z, w) (all variables take distinct values)
2z − w ≥ 0
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 39
![Page 221: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/221.jpg)
Constraint Programming Example
x ∈ {1, 3}y ∈ {1, 2, 3}z ∈ {1, 2, 3}w ∈ {1, 2, 4}
element(z, [1, 3, 5], x) (x is the zth element of [1, 3, 5])
alldifferent(x, y, z, w) (all variables take distinct values)
2z − w ≥ 0
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 39
![Page 222: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/222.jpg)
Constraint Programming Example
x ∈ {1, 3}y ∈ {1, 2, 3}z ∈ {1, 2, 3}w ∈ {1, 2, 4}
element(z, [1, 3, 5], x) (x is the zth element of [1, 3, 5])
alldifferent(x, y, z, w) (all variables take distinct values)
2z − w ≥ 0
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 39
![Page 223: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/223.jpg)
Constraint Programming Example
x ∈ {1, 3}y ∈ {1, 2, 3}z ∈ {1, 2, 3}w ∈ {1, 2, 4}
element(z, [1, 3, 5], x) (x is the zth element of [1, 3, 5])
alldifferent(x, y, z, w) (all variables take distinct values)
2z − w ≥ 0
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 39
![Page 224: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/224.jpg)
Constraint Programming Example
x ∈ {1, 3}y ∈ {1, 2, 3}z ∈ {1, 2, 3}w ∈ {1, 2, 4}
element(z, [1, 3, 5], x) (x is the zth element of [1, 3, 5])
alldifferent(x, y, z, w) (all variables take distinct values)
2z − w ≥ 0
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 39
![Page 225: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/225.jpg)
Constraint Programming Example
x ∈ {1, 3}y ∈ {1, 2, 3}z ∈ {1, 2, 3}w ∈ {1, 2, 4}
element(z, [1, 3, 5], x) (x is the zth element of [1, 3, 5])
alldifferent(x, y, z, w) (all variables take distinct values)
2z − w ≥ 0
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 39
![Page 226: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/226.jpg)
Constraint Programming Example
x ∈ {1, 3}y ∈ {1, 2, 3}z ∈ {1, 2, 3}w ∈ {1, 2, 4}
element(z, [1, 3, 5], x) (x is the zth element of [1, 3, 5])
alldifferent(x, y, z, w) (all variables take distinct values)
2z − w ≥ 0
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 39
![Page 227: An Integrated Solver for Optimization Problemscoral.ie.lehigh.edu/mip-2006/talks/Yunes.pdf · An Integrated Solver for Optimization Problems Ionut¸ D. Aron1 John N. Hooker2 Tallys](https://reader030.vdocuments.mx/reader030/viewer/2022040201/5e4a655ac095175eca7d7365/html5/thumbnails/227.jpg)
Constraint Programming Example
x ∈ {1, 3}y ∈ {1, 2, 3}z ∈ {1, 2, 3}w ∈ {1, 2, 4}
element(z, [1, 3, 5], x) (x is the zth element of [1, 3, 5])
alldifferent(x, y, z, w) (all variables take distinct values)
2z − w ≥ 0
Aron, Hooker and Yunes An Integrated Solver for Optimization Problems 39